Amenable crossed product Banach algebras associated with a class ...

arXiv:1705.02623v1 [math.FA] 7 May 2017

AMENABLE CROSSED PRODUCT BANACH ALGEBRAS ASSOCIATED WITH A CLASS OF C∗ -DYNAMICAL SYSTEMS. II MARCEL DE JEU, RACHID EL HARTI, AND PAULO R. PINTO A BSTRACT. We prove that the crossed product Banach algebra ℓ1 (G, A; α ) that is associated with a C∗ -dynamical system (A, G, α ) is amenable if G is a discrete amenable group and A is a strongly amenable C∗ -algebra. This is a consequence of the combination of a more general result with Paterson’s characterisation of strongly amenable unital C∗ -algebras in terms of invariant means for their unitary groups.

1. I NTRODUCTION

AND OVERVIEW

If (A, G, α ) is a C∗-dynamical system, where A is a nuclear C∗-algebra and G is an amenable locally compact Hausdorff topological group, then the crossed product C∗-algebra A ⋊α G is a nuclear C∗-algebra; see e.g. [10], [11, Proposition 14], or [24, Theorem 7.18]. Using Connes’ work in [5] and Haagerup’s in [12], one can equivalently say that A ⋊α G is an amenable Banach algebra if G is an amenable locally compact Hausdorff topological group and A is an amenable C∗-algebra. Here a Banach algebra A is called amenable if every bounded derivation of A with values in a dual Banach Abimodule is inner, and a topological group G is called amenable if there exists a left invariant state on the unital C∗-algebra of bounded right uniformly continuous complex valued functions on G. For a locally compact Hausdorff topological group G, this is equivalent to the existence of left invariant states on other unital C∗-algebras of (equivalence classes of) functions on G; see e.g. [23, Definition 1.1.4, Theorem 1.1.9, and Theorem 1.1.11]. If G is discrete, amenability is simply the existence of a left invariant state on ℓ∞ (G). The C∗-algebra A ⋊α G is the enveloping C∗-algebra of the twisted convolution algebra L1 (G, A; α ). In view of the above reformulation of the nuclearity result in terms of amenability, it seems natural to inquire whether L1 (G, A; α ) is perhaps also amenable under the same conditions on A and G. 2010 Mathematics Subject Classification. Primary 47L65; Secondary 43A07, 46H25, 46L55. Key words and phrases. Crossed product Banach algebra, C∗ -dynamical system, amenable Banach algebra, amenable group, strongly amenable C∗ -algebra. 1

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MARCEL DE JEU, RACHID EL HARTI, AND PAULO R. PINTO

Apart from its intrinsic interest, this would also provide an alternative approach to the nuclearity of A ⋊α G. Indeed, since the inclusion of the latter in the former is continuous (even contractive) with dense image, we could then use [23, Proposition 2.3.1] to conclude that A ⋊α G is amenable, and therefore nuclear. Not much seems to be known about the amenability of L1 (G, A; α ), or, for that matter, of other Banach algebras of L1 -type at all. There is, of course, Johnson’s result for A = : if G is an amenable locally compact Hausdorff topological group, then L1 (G) is amenable. See e.g [13, Theorem 2.5] for this and its converse; the latter is due to Ringrose. If G is discrete, then a little more is known. The algebra ℓ1 (G, A; α ) (we shall give its definition in Section 2) is amenable if A is a commutative or finite dimensional C∗algebra (see [6, Theorem 2.4]); it is unknown whether there is a converse of some kind involving properties of G. For general G and (Banach) algebra A, one can introduce a weight ω : G → ≥0 and arrive at a generalised Beurling algebra L1 (G, A, ω ; α ) as in [7, Definition 5.4]. For G = and A = , it is known (see [1, Theorem 2.4]) that for certain weights the ensuing Beurling algebras are amenable (or weakly amenable), whereas for others they are not. The authors are not aware of any other results on amenability for L1 -type Banach algebras. In this paper, we show that ℓ1 (G, A; α ) is an amenable Banach algebra if G is a discrete amenable group and A is a strongly amenable not necessarily unital C∗-algebra (see Theorem 4.3). The latter notion has been introduced by Johnson (see [13, p.70] or [12, p. 313]): a unital C∗-algebra with unitary group U is said to be strongly amenable if, for every bounded derivation D of A with values in a dual Banach A-bimodule E ∗ , there exists x∗ in the weak*-closed convex hull of { −Du · u−1 : u ∈ U } in E ∗ such that Da = a · x − x · a for all a ∈ A. A non-unital C∗-algebra is said to be strongly amenable if its unitisation is. Every strongly amenable Banach algebra is amenable, but the converse does not hold, as is shown by the Cuntz algebras On for n ≥ 2; see [21]. All Type I C∗-algebras (equivalently: all postliminal C∗-algebras; see [8, Remark 9.5.9]) are strongly amenable; see [13, Theorem 7.9]. Thus our present result covers a reasonably wide class of examples, and, in particular, it implies our previous result that ℓ1(G, A; α ) is amenable if A is a commutative or finite dimensional C∗-algebra. We shall now explain the structure of the proof, which will also make clear how strong amenability of A enters the picture in a natural way, replacing the weaker requirement of amenability that was the initial Ansatz in the above discussion. Actually, it will become clear which (presumably) weaker condition than strong amenability is sufficient for ℓ1 (G, A; α ) to be amenable.

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AMENABLE CROSSED PRODUCT BANACH ALGEBRAS

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Let us sketch how one could attempt (and fail) to prove—along the lines of [6]—that ℓ1 (G, A; α ) is an amenable Banach algebra if G is a discrete amenable group and A is only known to be an amenable C∗-algebra. First of all, it follows from [23, Corollary 2.3.11] and [6, Lemma 2.2] that it is equivalent to attempt this with A also unital, so let us assume this. In that case (we refer to Lemma 2.1 for details), ℓ1 (G, A; α ) contains a group H that is generated as an abstract group by G and the unitary group U of A, and that has U as a normal subgroup. In fact, the group H is isomorphic to U ⋊α G as abstract groups, but we shall not need this more precise statement. The important point is that the closed linear span of H equals ℓ1 (G, A; α ). Therefore, if D is a bounded derivation of ℓ1 (G, A; α ) with values in a dual ℓ1 (G, A; α )-bimodule E ∗ , and if we want to show that D is inner, we need only prove that its restriction to H is inner. This observation is already used in [6]. In [6], the proof then proceeds by supplying U with the inherited norm topology of ℓ1(G, A; α ) if A is finite dimensional, or with the discrete topology if A is commutative. Then U is an amenable locally compact Hausdorff topological group in both cases, and a known stability property for such groups (see [18, Proposition 13.4]) then implies that H is an amenable locally compact Hausdorff topological group as well. Consequently, [18, Theorem 11.8.(ii)] (see also [18, p. 17–18 and p. 99]) shows that D is inner on H. This concludes the proof in [6]. Inspection shows that the proof of the result on innerness that is invoked (i.e. of [18, Theorem 11.8.(ii)]) is ultimately based (see [18, proof of Lemma 11.6]) on Johnson’s archetypical argument (see [13, p. 33]) to show that, in a suitable context, bounded derivations of a group with values in dual Banach bimodules are inner if there exists an appropriate left invariant mean on the group. The same is thus true for our earlier result [6, Theorem 2.4]: surviving all layers if A is commutative or finite dimensional, it is this argument that provides the key to the amenability of ℓ1 (G, A; α ) in [6]. For general amenable A, the natural follow-up after the observation that one need only prove that D is inner on H does not seem to work. We shall now explain this. For a general unital C∗-algebra A, it has been established by Paterson (see [17, Theorem 2]) that its unitary group U is a Hausdorff topological group in the weak topology that it inherits from the Banach space A, and that U is amenable in that weak topology if A is amenable (in fact, this characterises amenable A; see [17, Theorem 2]). In view of what has worked earlier this is an encouraging starting point, since we are indeed in that situation. So let us supply U with the weak topology inherited from A, and, for convenience, let us assume—this could perhaps be another matter—that

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we can show that H ≃ U ⋊α G is a topological group in the product topology, and that it is then amenable. In that case, one has a left H-invariant mean on the bounded right uniformly continuous functions on H to work with, and the next step would presumably be to use Johnson’s argument to show that this implies that a bounded derivation on H with values in a dual Banach H-bimodule E ∗ is inner. As earlier, this would then conclude the proof. It is at this point, however, that an obstacle arises. In order to be able to apply Johnson’s argument to H, one needs that, for all x ∈ E, the function h 7→ hx, Dh · h−1 i is in the function space on which the left invariant mean living on H acts. In the presumed situation, it should, therefore, be bounded and right uniformly continuous. In particular, its restriction to U should be right uniformly continuous. However, there seems to be no reason why this should in general be the case or, for that matter, why it should even be continuous. If the actions of U on E are strongly continuous, then these restricted functions are easily seen to be right uniformly continuous, as required, but we have no guarantee that this is the case if the U -actions and the derivation D of U originate from an enveloping ℓ1 (G, A; α )-bimodule structure. The point is that we are working with the group U in its inherited weak topology from A, and not in the inherited norm topology from A. In the latter topology the actions of U on E are evidently strongly continuous, but there seems to be no reason why this should still be the case for the weak topology on U , and it is the latter topology we must work with if we want to have the amenability of U from Paterson’s result at our disposal. Thus this attempt, based on combining Paterson’s result for amenable unital C∗-algebras and Johnson’s argument exploiting a left invariant mean on the right uniformly continuous functions on a topological group, runs aground. It is at this point that Paterson’s characterisation of strongly amenable unital C∗-algebras (see [16, Theorem 2] and the left/right discussion preceding Theorem 4.3) comes in to overcome this obstruction originating from having the ‘wrong’ topology on U . Indeed, if A is strongly amenable, then there exists a left invariant mean on a space of functions on U that makes no reference to a specific topology on U at all, but that is naturally associated with the bounded bilinear forms on A. It is immediate (see the proof of Theorem 4.3) that the functions u 7→ hx, Du · u−1 i are in this space for all x ∈ E, together with the constants. It is then not too difficult—it is here that the amenability of the discrete group G is used; see the proof of Lemma 3.5)—to construct a left invariant mean on the minimal space of functions on H on which one needs such a mean in order to be able to make Johnson’s argument work, i.e. on the space of functions on H that is spanned by the constants and the functions h 7→ hx, Dh · h−1 i for x ∈ E (see the proof of Lemma 3.2). With this available, D is seen to be inner on H, and then we are done. It is in this way, by combining Paterson’s result [16, Theorem 2]


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for strongly amenable unital C∗-algebras with a somewhat more careful inspection of what the minimal requirements are in order to be able to apply Johnson’s argument, that the proof is then concluded for such algebras after all. As will become apparent from Section 3, these minimal requirements are not related to topology at all. In the end, therefore, there is no role anymore for the amenability of U in the inherited weak topology of A, even though at first this seemed to be the most natural thing to start with. Actually, as may already be obvious from the above discussion, one does not really need that A is strongly amenable for ℓ1 (G, A; α ) to be amenable. The existence of a left invariant mean on the space of functions on U that is spanned by the constants and the functions u 7→ hx, Du · u−1i for x ∈ E is already sufficient, and Paterson’s result [16, Theorem 2] ‘only’ implies that this is certainly the case if A is strongly amenable. It is, therefore, perhaps more precise to regard Theorem 4.2, which is still in the general context, as the most prominent result of this paper. The fact that ℓ1(G, A; α ) is amenable if A is strongly amenable and G is amenable is an appealing special case thereof. This paper is organised as follows. Section 2 contains the necessary terminology and definitions, including that of the Banach algebra ℓ1 (G, A; α ). Section 3 is concerned with prudent hypotheses implying that an abstract group has the property that every bounded derivation with values in a dual Banach bimodule is inner. Section 4 contains the main results Theorems 4.2 and 4.3 on the amenability of ℓ1 (G, A; α ). In Section 5 we briefly discuss converse implications. The situation here is largely open, with only a limited number of results available that can be derived via the detour of the enveloping C∗-algebra A ⋊α G of ℓ1 (G, A; α ). It is also argued here that this detour could involve loss of information, so that proper L1 -type arguments are needed. For a discussion of a surmised general framework for amenability of crossed products of Banach algebras (as in [7, Definition 3.2]) that are associated with C∗-dynamical systems we refer to [6, Section 3]. 2. P RELIMINARIES We start by establishing some terminology and notation. Since the definitions in the literature can slightly vary from author to author (or even greatly in the case of ‘left’ and ‘right’), we shall also define the most basic notions. If G is an abstract group, then ℓ∞ (G) denotes the space of all bounded complex valued functions on G. All subspaces of ℓ∞ (G) will be assumed to g carry the supremum norm. For g0 ∈ G, define the left translation LG0 : ℓ∞ (G) →

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ℓ∞(G) by LgG0 ϕ (g) = ϕ (g0 g) for ϕ ∈ ℓ∞(G) and g ∈ G. Then LgG1 g2 = LgG2 LgG1 for g1 , g2 ∈ G. The definition of a right translations is similar, without an inverse. We have included the group in the notation, as later on there will be several groups occurring simultaneously. If ΦG ⊂ ℓ∞(G) is a not necessarily closed subspace, then ΦG is said to g be left G-invariant if it is invariant under LG for all g ∈ G. An element x∗Φ G of the norm dual ΦG∗ of a left G-invariant subspace ΦG of ℓ∞ (G) is said to be left G-invariant if hLgG ϕ , x∗Φ i = hϕ , x∗Φ i for all ϕ ∈ ΦG and g ∈ G. A G G right G-invariant subspace and a right G-invariant element of its dual are similarly defined. Although we have employed it in Section 1, we shall not use the terminology of ‘means’ in the sequel, but simply state the properties that an element m of the norm dual of a subspace of ℓ∞ (G) is required to have. As we shall see, the state-like property that kmk = 1 = m(1G ) (here 1G denotes the function on G that is identically 1) will never be needed; see, however, part 2 of Remark 3.3. If G is an abstract group, then a Banach left G-module is a Banach space E that is supplied with a left G-action with the property that there exists K ≥ 0 such that kg · xk ≤ Kkxk for all g ∈ G an x ∈ E. We do not assume that the G-action is unital. The definitions of a Banach right G-module and of a Banach G-bimodule are analogous. A dual Banach G-bimodule is obtained from a Banach G-bimodule by taking the adjoint actions. If G is an abstract group, and E is a Banach G-bimodule, then a derivation of the group G with values in E is a map D : G 7→ E such that D(g1 g2 ) = Dg1 · g2 + g1 · Dg2 for all g1 , g2 ∈ G. It is a bounded derivation of the group G if D(G) is a norm bounded subset of E. For x ∈ E, the map g 7→ g · x − x · g for g ∈ G is a bounded derivation of G; such a derivation is called an inner derivation of the group G If A is a Banach algebra, then a Banach left A-module is a Banach space E that is supplied with a left A-action with the property that there exists a constant K ≥ 0 such that ka · xk ≤ Kkakkxk for all a ∈ A and x ∈ E. We do not assume that (if applicable) the A-action is unital. The definitions of a Banach right A-module and a Banach A-bimodule are analogous. A dual Banach A-bimodule is obtained from a Banach A-bimodule by taking the adjoint actions. If A is a Banach algebra, and E is a Banach A-bimodule, then a derivation of the Banach algebra A with values in E is a linear map D : A 7→ E such that D(a1 a2 ) = Da1 · a2 + a1 · Da2 for all a1 , a2 ∈ A. It is a bounded derivation of the Banach algebra A if D is a bounded operator between the normed spaces A and E. For x ∈ E, the map a 7→ a · x − x · a for a ∈ A is a bounded derivation of A. Such a derivation is called an inner derivation of the Banach


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algebra A. The Banach algebra A is an amenable Banach algebra if every bounded derivation of A with values in a dual Banach A-bimodule is an inner derivation. After this part on terminology and notation, we turn to the description of the Banach algebra ℓ1 (G, A; α ). Let (A, G, α ) be a C∗-dynamical system with G discrete, i.e. let α : G → Aut(A) be an action of a discrete group G as ∗ -automorphisms of a (not necessarily unital) C∗-algebra A. Let ℓ1 (G, A; α ) = { a : G −→ A : kak :=

∑ kagk < ∞ },

g∈G

where for typographical reasons we have written ag for a(g). We supply ℓ1 (G, A; α ) with the usual twisted convolution product and involution, defined by (2.1)

(aa′ )(g) =

∑ ah · αh(a′h−1g) h∈G

for a, a′ ∈ ℓ1 (G, A; α )) and g ∈ G, and by (2.2)

a∗ (g) = αg ((ag−1 )∗ )

for a ∈ ℓ1 (G, A; α )) and g ∈ G, respectively. Then ℓ1 (G, A; α ) becomes a Banach algebra with isometric involution. The usual convolution algebra ℓ1 (G, A) is the special case ℓ1 (G, A; triv). Specialising further, if A = C, then ℓ1 (G, C; triv) is the usual group algebra ℓ1 (G). Suppose that A is unital. In that case, there is a more convenient model for ℓ1 (G, A; α ), as we shall now indicate. For g ∈ G, let δg : G → A be defined by 1A if h = g; δg (h) = 0A if h 6= g, where 1A and 0A denote the identity and the zero element of A, respectively. Then δg ∈ ℓ1 (G, A; α ) and kδg k = 1 for all g ∈ G. Furthermore, ℓ1(G, A; α ) is unital with δe as identity element, where e denotes the identity element of G. Using equation (2.1), one finds that

δgh = δg · δh for all g, k ∈ G. Hence δg is invertible in ℓ1(G, A; α ) for all g ∈ G, and we have δg−1 = δg−1 . It is now obvious that the set { δg : g ∈ G } consists of norm one elements of ℓ1 (G, A; α ), and that it is a subgroup of the invertible elements of ℓ1 (G, A; α ) that is isomorphic to G. In the same vein, it follows easily from equations (2.1) and (2.2) that we can view A isometrically as a closed *-subalgebra of ℓ1 (G, A; α ), namely

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as { aδe : a ∈ A }, where aδe is the element of ℓ1 (G, A; α ) that assumes the value a ∈ A at e ∈ G, and the value 0A ∈ A elsewhere. If a ∈ ℓ1 (G, A; α ), then it is easy to see that a = ∑g∈G (ag δe )δg as an absolutely convergent series in ℓ1 (G, A; α ). Hence, if we identify ag δe and ag , we have a = ∑g∈G ag δg as an absolutely convergent series in ℓ1 (G, A; α ). Finally, let us note that an elementary computation, using equation (2.1) and the identifications just mentioned, shows that the identity

δg aδg−1 = αg (a) holds in ℓ1 (G, A; α ) for all g ∈ G and a ∈ A. The following key observation, already used in [6], is now clear. Lemma 2.1. Let (A, G, α ) be a C∗-dynamical system, where G is a discrete group and A a unital C∗-algebra with unitary group U . Let H = { uδg : u ∈ U, g ∈ G }. Then H and the semidirect product U ⋊α G are canonically isomorphic as abstract groups. Moreover, the linear span of H is dense in ℓ1 (G, A; α ). 3. I NNERNESS

OF DERIVATIONS OF ABSTRACT GROUPS

This section is concerned with prudent hypotheses ensuring that a bounded derivation of an abstract group G with values in a dual Banach G-bimodule is inner. We start by employing Johnson’s argument [13, p. 33] under such hypotheses. It seems customary in the literature to first change the G-bimodule structure into one where the left action is trivial, prove that for such Gbimodule structures bounded derivations in dual Banach G-modules are inner, and then conclude from this special case that this also holds in the general case. As the proof of Lemma 3.2 below shows, this is hardly an actual simplification. It also shows how the map g 7→ Dg · g−1 , occurring in the definition of a strongly amenable unital C∗-algebra, is quite natural in a more general context. In the proof, a family of functions occur that will frequently reappear in the sequel. We shall now define these, and subsequently proceed with the innerness of derivations. Definition 3.1. Let G be an abstract group, and let E be a Banach Gbimodule. Let D : G → E ∗ be a derivation. For x ∈ E, define ϕGx,D : G 7→ by ϕGx,D (g) = hx, Dg · g−1i for g ∈ G.

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Lemma 3.2. Let G be an abstract group, and let E be a Banach G-bimodule where the left G-action on E is unital.


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Let D : G → E ∗ be a bounded derivation. Then ϕGx,D ∈ ℓ∞ (G) for all x ∈ E. Put x,D ΦD G = span { { ϕG : x ∈ E } ∪ { 1G } }. Then ΦD G is left G-invariant. ∗ Suppose that there exists a left G-invariant element mΦD of ΦD with G G h1G , mΦD i = 1. Then D is inner. G

Proof. It is clear that ϕGx,D ∈ ℓ∞ (G) for all x ∈ E, so we start with the left G-invariance of ΦD G . Let x ∈ E and g0 ∈ G. Then, for g ∈ G, g

(LG0 ϕGx,D )(g) = ϕGx,D (g0 g) = hx, D(g0g) · (g0g)−1 i −1 −1 = hx, Dg0 · g · g−1 · g−1 · g0 i 0 + g0 · Dg · g −1 −1 = hx, Dg0 · g−1 0 i + hg0 · x · g0 , Dg · g i.

Hence (3.1)

g−1 ·x·g0 ,D

0 LgG0 ϕGx,D = hx, Dg0 · g−1 0 i1G + ϕG

for all x ∈ E and g ∈ G. Fromthis the left G-invariance of ΦD G is clear. ∗ D If an element mΦD of ΦG as specified exists, then define x∗0 : E → G by hx, x∗0 i = hϕGx,D , mΦD i

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G

for x ∈ E. The boundedness of mΦD , of D, and of the left G-action on E G ensure that x∗0 is bounded. Replacing g0 with g, and x with g · x in equation (3.1), we find that g

g·x,D

LG ϕG

x·g,D

= hx, Dgi1G + ϕG

for all x ∈ E and g ∈ G; it is here that we use that the left G-action on E is unital. Using this, we see that, for all x ∈ E and g ∈ G, hx, g · x∗0 i = hx · g, x∗0 i x·g,D

= hϕG = = = =

, mΦD i G

hLgG ϕGg·x,D − hx, Dgi1G , mΦD i G g·x.D hϕG , mΦD i − hx, Dgi G ∗ hg · x, x0 i − hx, Dgi hx, x∗0 · gi − hx, Dgi.

Hence Dg = x∗0 · g − g · x∗0 for all g ∈ G, so that D is inner.

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Remark 3.3. (1) Note that it is not required in Lemma 3.2 that kmΦD k = 1, i.e. we do G not impose the state-like condition kmΦD k = 1 = h1G , mΦD i. G G (2) Suppose that there exists a left G-invariant element mℓ∞ (G) of ℓ∞(G)∗ such that h1G , mℓ∞ (G) i = 1. In that case, Lemma 3.2 shows that every bounded derivation of G with values in an arbitrary dual Banach G-bimodule E ∗ is inner, provided that the left G-action on E is unital. Then [18, Theorem 11.8.i] implies that the discrete group G is amenable in the sense of [18, Definition 4.2] (see also [18, Definition 3.1]), and consequently [18, Proposition 3.2] yields that there exists a left G-invariant state on ℓ∞ (G). Thus the discrete group G is, in fact, an amenable topological group as this notion was defined in Section 1. However, in order to emphasise that it is the ‘actual’ natural condition, we shall, when applicable, insist on using the seemingly weaker requirement that there exist a left G-invariant element mℓ∞ (G) of ℓ∞ (G)∗ such that h1G , mℓ∞ (G) i = 1. Remark 3.4. Inspection of the proof of Lemma 3.2 shows that the hypotheses on the G-actions and D can be relaxed. It is sufficient that the left and right G-actions on E be by bounded operators, that the left G-action on E be unital, and that { Dg · g−1 : g ∈ G } be a norm bounded subset of E ∗. The next step is to investigate the innerness of bounded derivations of a group that is generated by two subgroups, one of which is normal. For this result, Proposition 3.6 below, we need the following preparation. Lemma 3.5. Let G be an abstract group that is generated by a subgroup H and a normal subgroup N. Let ΦN be a left N-invariant subspace of ℓ∞ (N) containing the constants, and let ΦG be a left G-invariant subspace of ℓ∞ (G) containing the constants and such that ϕ ↾N ∈ ΦN for all ϕ ∈ ΦG . Suppose that there exists a left H-invariant element mℓ∞ (H) of ℓ∞ (H)∗ with h1H , mℓ∞ (H) i = 1, as well as a left N-invariant element mΦN of ΦN∗ with h1N , mΦN i = 1. Then there exists a left G-invariant element mΦG of ΦG∗ with h1G , mΦG i = 1. Every g ∈ G can be written as g = hn with h ∈ H and n ∈ N, but it is not required that this factorisation be unique, i.e. it is sufficient that G be a quotient of an external semi-direct product of its subgroups H and N, and not necessarily equal to an internal semi-direct product of these subgroups. When applying Lemma 3.5 in our principal case of interest (see the proof


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of Proposition 4.1) the latter will actually be the case, but it plays no role in the proofs. Proof. This is a variation on a construction that is usually carried out in the context of a discrete (see e.g. [3, Proposition 4.5.5]) or, more generally (see e.g. [18, Proposition 13.4]), a locally compact Hausdorff topological group G having a normal closed subgroup N such that N and G/N are both amenable. For ϕ ∈ ΦG , let ϕ˜ ∈ ℓ∞ (H) be defined by

ϕ˜ (h) = h(LhG ϕ )↾N , mΦN i, for h ∈ H, and put hϕ , mΦG i = hϕ˜ , mℓ∞ (H) i. Then mΦG ∈ Φ∗G , and h1G , mΦG i = 1.

It is elementary that (LhG0 ϕ )∼ = LhH0 ϕ˜ for all ϕ ∈ ΦG and h0 ∈ H. The LhH0 -invariance of mℓ∞ (H) then implies that mΦG is invariant under LhG for all h ∈ H. For ϕ ∈ ΦG , n0 ∈ N, and h0 ∈ H fixed, we have (LnG0 ϕ )∼ (h0 ) = h(LhG0 LnG0 ϕ )↾N , mΦN i = h(LnG0h0 ϕ )↾N , mΦN i.

Since N is normal, there exists n′0 ∈ N such that n0 h0 = h0 n′0 , and it is eleh n′

n′

n′

mentary that (LG0 0 ϕ )↾N = LN0 [(LhG0 ϕ )↾N ]. Hence, using the LN0 -invariance of mΦN in the penultimate step, we have (LnG0 ϕ )∼ (h0 ) = h(LnG0 h0 ϕ )↾N , mΦN i h n′

= h(LG0 0 ϕ )↾N , mΦN i n′

= hLN0 [(LhG0 ϕ )↾N ], mΦN i = h(LhG0 ϕ )↾N , mΦN i = ϕ˜ (h0 ). Therefore (LnG0 ϕ )∼ = f˜; it follows that mΦG is also LnG -invariant for all n ∈ N. Since G is generated by H and N, this concludes the proof. We can now combine Lemma 3.2 and Lemma 3.5 and obtain the following result. The prudent formulation of the hypotheses in Lemmas 3.2 and 3.5 allows us to be likewise prudent in the hypotheses on the group N in the following result. The functions ϕNx,D figuring in it are the ones as in

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Definition 3.1, but then for the restrictions of the actions and of the derivation to N. Obviously, these are simply the restrictions of the ϕGx,D to N. Proposition 3.6. Let G be an abstract group that is generated by a subgroup H and a normal subgroup N, where H has the property that there exists a left H-invariant element mℓ∞ (H) of ℓ∞ (H)∗ with h1H , mℓ∞ (H) i = 1. Let E be a Banach G-bimodule where the left G-action on E is unital, and let D : G → E ∗ be a bounded derivation. Let the left N-invariant subspace ∞ ΦD N of ℓ (N) be defined by x,D ΦD N = span { { ϕN : x ∈ E } ∪ { 1N } },

and suppose that there exists a left N-invariant element mΦD of ΦD N N h1N , mΦD i = 1. N Then D is an inner derivation of G.

∗

with

Proof. Put x,D ΦD G = span { { ϕG : x ∈ E } ∪ { 1G } }.

The first part of Lemma 3.2 shows that ΦD N is a left N-invariant subspace of ℓ∞ (N), and that ΦD is left G-invariant subspace of ℓ∞ (G). From the G comments preceding the proposition, it is trivial that the set of restrictions D of the elements of ΦD G to N equals ΦN . We are now in the situation of Lemma 3.5,and we conclude that there ∗ exists a left G-invariant element mΦD of ΦD with h1G , mΦD i = 1. An G G G appeal to Lemma 3.2 shows that D is an inner derivation of G. Remark 3.7. In view of Remark 3.4 and the above proof, the hypotheses in Proposition 3.6 on the G-actions and D can be relaxed. It is sufficient that the left and right G-actions on E be by bounded operators, that the left G-action on E be unital, and that { Dg · g−1 : g ∈ G } be a norm bounded subset of E ∗ . 4. A MENABILITY

OF ℓ1 (G, A; α )

All that remains to be done is to combine Lemma 2.1 and Proposition 3.6, and add Paterson’s result [16, Theorem 2] later on. This is made possible by the prudence concerning hypotheses in Section 3. We recall that the condition on G occurring in the results in this section is actually equivalent to requiring that there exist a left G-invariant state on ℓ∞(G); see part 2 of Remark 3.3 The following result is the most precise one in this section, because it is concerned with only one bounded derivation of ℓ1 (G, A; α ). For the convenience of the reader, the definition of the functions ϕUx,D from Definition 3.1 is included again, as it will be in Theorem 4.2.


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Proposition 4.1. Let (A, G, α ) be a C∗-dynamical system, where A is a unital C∗-algebra with unitary group U , and G is a discrete group such that there exists a left G-invariant element mℓ∞ (G) of ℓ∞ (G)∗ with h1G , mℓ∞ (G) i = 1. Let E be a Banach ℓ1 (G, A; α )-bimodule, and let D : ℓ1 (G, A; α ) → E ∗ be a bounded derivation. For x ∈ E, define ϕUx,D : U 7→ by

C

ϕUx,D (u) = hx, Du · u−1i for u ∈ U , and put D ΦU = span { { ϕUx,D : x ∈ E } ∪ { 1U } }. D Suppose that there exists a left U -invariant element mΦD of ΦU U h1U , mΦD i = 1. U Then D is an inner derivation of ℓ1 (G, A; α ).

∗

with

Proof. Assume first that E is unital as a left ℓ1 (G, A; α )-module. Consider H ≃ U ⋊α G ⊂ ℓ1(G, A; α ). Obviously, E is a H-bimodule. Since H is a bounded subset of ℓ1 (G, A; α ) (recall that khk = 1 for all h ∈ H), E is, in fact, a Banach H-bimodule. Since E is unital as a left ℓ1 (G, A; α )-module, and the identity elements of ℓ1 (G, A; α ) and H coincide, the left H-action on E is unital. The derivation D of ℓ1 (G, A; α ) restricts to a derivation of H, again denoted by D. Since H is a bounded subset of ℓ1 (G, A; α ), D is a bounded derivation of U with values in E ∗ . We can now apply Proposition 3.6, where we replace G with our present H, H with our present G, and N with our present U . We conclude from Proposition 3.6 that D is inner on our present H. Since by Lemma 2.1 the linear span of our present H is dense in ℓ1 (G, A; α ), D is inner on ℓ1 (G, A; α ). This concludes the proof where E is unital as a left ℓ1 (G, A; α )-module. It remains to cover the case where E is possibly degenerate as a left ℓ1 (G, A; α )-module. The approach for this is more or less standard (cf. e.g. [13, Proposition 1.8] or [19, Proposition 0.3]), but since our criterion applies to one derivation, and is in terms of a specific space of functions on U that is defined using the whole space E, we cannot content ourselves with a reference to a known isomorphism between two cohomology groups, one of which then corresponds to a unital left action on the largest submodule with this property. Hence we include the required steps. Denote the left action of the identity element δe of ℓ1 (G, A; α ) on E by P. Then P is a continuous projection with adjoint P∗ : E ∗ → E ∗ . Both are morphisms of ℓ1 (G, A; α )bimodules. The subspace PE is a Banach ℓ1 (G, A; α )-bimodule such that the left action of ℓ1 (G, A; α ) is unital. In order to describe the dual Banach ℓ1 (G, A; α )-bimodule (PE)∗ of PE, define j : P∗ E ∗ → (PE)∗ by j(x∗ ) =

14


x∗ ↾PE for x∗ ∈ P∗ E ∗ . Then j is a topological isomorphism of Banach ℓ1 (G, A; α )-bimodules between P∗ E ∗ and (PE)∗ . Define DP : ℓ1 (G, A; α ) → (PE)∗ by j(a) = j(P∗ Da) for a ∈ A. Then DP is a bounded derivation of ℓ1 (G, A; α ) with values in the dual Banach ℓ1 (G, A; α )-bimodule (PE)∗ . DP D . From the Moreover, it is easily verified (this is the point) that ΦU = ΦU result for the left unital case, we know that there exists ξ ∗ ∈ (PE)∗ such that DP a = a · ξ ∗ − ξ ∗ · a for all a ∈ ℓ1 (G, A; α ). It is then a routine further verification that D is the inner derivation of ℓ1 (G, A; α ) that corresponds to j−1 ξ + (idE ∗ − P∗ )Dδe ∈ E ∗ . In the unital case, it is now obvious what the natural sufficient condition is for ℓ1 (G, A; α ) to be amenable. The following is immediate from Proposition 4.1. Theorem 4.2. Let (A, G, α ) be a C∗-dynamical system, where A is a unital C∗-algebra with unitary group U , and where G is a discrete group such that there exists a left G-invariant element mℓ∞ (G) of ℓ∞ (G)∗ with h1G , mℓ∞ (G) i = 1. For every Banach ℓ1 (G, A; α )-bimodule E, every x ∈ E, and every bounded derivation D : ℓ1 (G, A; α ) → E ∗ , define ϕUx,D : U 7→ by

C

ϕUx,D (u) = hx, Du · u−1i for u ∈ U . Der denote the left U -invariant subspace of ℓ∞ (U ) that is spanned by Let ΦU 1U and the functions ϕUx,D corresponding to all Banach ℓ1 (G, A; α )-bimodules E, all x ∈ E, and all bounded derivations D : ℓ1 (G, A; α ) → E ∗ . Der ∗ with Suppose that there exists a left U -invariant element mΦDer of ΦU U

h1U , mΦU i = 1. Then ℓ1 (G, A; α ) is an amenable Banach algebra.

It remains to identify a substantial class of unital C∗-algebras such that mΦDer as in Theorem 4.2 exists. As we shall now proceed to show, the U strongly amenable C∗-algebras are such a class. Let A be a unital C∗-algebra with unitary group U . Let Bil(A) be the space of all bounded bilinear forms on A. Following [16], for V ∈ Bil(A), we Bil = span { ∆V : V ∈ Bil(A) }; define ∆(V ) : U → by ∆(V )(u) = V (u−1 , u). Let ΦU Bil is a subspace of ℓ∞ (U ) this space is denoted by B(A) in [16]. Then ΦU containing the constants (consider (a1 , a2 ) 7→ a∗ (a1 a2 ) for a∗ ∈ A∗ such that a∗ (1A ) = 1), and it is easy to see that it is left and right U -invariant; see [16, p. 557]. It can then be shown (see [16, Theorem 2]) that A is strongly amenable if and only if there exists a right U -invariant element Bil ∗ such that m mΦBil of ΦU ΦBil (1) = 1 = kmΦBil k.

C

U

U

U


15

Actually, the strong amenability of a unital C∗-algebra A is also equivalent to the existence ∗ of a left (which is what we need) U -invariant element Bil such that mΦBil (1) = 1 = kmΦBil k. We shall now show mΦBil of ΦU U U U this. For ϕ ∈ ℓ∞ (U ), define ϕˇ ∈ ℓ∞ (U ) by ϕ (u) = ϕ (u−1 ) for u ∈ U . For V ∈ Bil(A), define Vˇ ∈ Bil(A) by Vˇ (a1 , a2 ) = V (a2 , a1) for a1 , a2 ∈ A. Then (∆V )ˇ = ∆(Vˇ ) for V ∈ Bil(A). We see from this that the map ϕ 7→ ϕˇ is an Bil and this, in turn, isometric linear automorphism of the normed space ΦU shows how to obtain left U -invariant continuous linear functionals from right U -invariant ones, and vice versa. With Theorem 4.2 and the above discussion of Paterson’s result available, the rest is now easy. Theorem 4.3. Let (A, G, α ) be a C∗-dynamical system, where A is a strongly amenable not necessarily unital C∗-algebra, and G is a discrete group such that there exists a left G-invariant element mℓ∞ (G) of ℓ∞ (G)∗ with h1G , mℓ∞ (G) i = 1. Then ℓ1 (G, A; α ) is an amenable Banach algebra. Proof. As a consequence of [6, Lemma 2.2] and the definition of strong amenability, we may assume that A is a strongly amenable unital C∗-algebra. In view of Theorem 4.2 and the discussion preceding the present theorem, Der ⊂ ΦBil . So let E be a Banach ℓ1 (G, A; α )we need only show that ΦU U bimodule, let x ∈ E, let D : ℓ1 (G, A; α ) → E ∗ be a bounded derivation, and consider the associated function ϕUx,D : U 7→ , defined by

C

ϕUx,D (u) = hx, Du · u−1i for u ∈ U . Define the bounded bilinear form V x,D on A by V x,D (a1 , a2 ) = ha1 · x, Da2 i for a1 , a2 ∈ A. Then ϕUx,D = ∆(V x,D ). It was already noted in Section 1 that all Type I (equivalently: all postliminal) C∗-algebras are strongly amenable. Therefore, Theorem 4.3 implies that ℓ1(G, A; α ) is amenable if A is a commutative or finite dimensional C∗algebra, which is [6, Theorem 2.4]. 5. C ONVERSES We shall now briefly discuss possible converse implications: if (A, G, α ) is a C∗-dynamical system where G is a discrete group, and if ℓ1 (G, A; α ) is an amenable Banach algebra, then what can one say about G or A? The following modest result sums up the results in this vein that the authors are aware of. Proposition 5.1. Let (G, A, α ) be a C∗-dynamical system where G is a discrete group. Suppose that ℓ1 (G, A; α ) is amenable. Then:

16


(1) The algebra A is amenable; (2) If A = , then G is an amenable group; (3) If A = C(X ) for a compact Hausdorff space X , then the action of G on X is amenable.

C

For the definition of an amenable G-action on X we refer to [2, Definition 4.3.5]. We note that it is not asserted in the first part that A is strongly amenable. In view of the logical dependence between Theorem 4.2 and Theorem 4.3 this is, perhaps, also not to be expected. After all, since—in the notation of Der ⊂ ΦBil could Section 4—there seems to be no reason why the inclusion ΦU U not be proper, one would, perhaps, expect that there are unital C∗-algebras that are not strongly amenable, but for which Theorem 4.2 still applies. On the other hand, in the realm of amenability of groups there are some surprising implications between the existence of invariant functionals on various function spaces, and we cannot entirely exclude the possibility that the conclusion in the first part of Proposition 5.1 can still be strengthened. Proof. We prove the first statement. The crossed product C∗-algebra A ⋊α G is the enveloping C∗-algebra of the involutive Banach algebra ℓ1 (G, A; α ). By the very construction of such an enveloping C∗-algebra, there is a continuous (even contractive) homomorphism of ℓ1 (G, A; α ) into A ⋊α G with dense range. Since ℓ1 (G, A; α ) is amenable, [23, Proposition 2.3.1] implies that A ⋊α G is amenable. Therefore, the reduced crossed product A ⋊α ,r G, which is a quotient of A ⋊α G, is also amenable. Since there exists a conditional expectation from A ⋊α ,r G onto A (see e.g. [2, Proposition 4.1.9]), this implies that A is amenable (see e.g. [2, Exercise 2.3.3]). The second statement is, of course, just a special case of Ringrose’s converse to Johnson’s theorem; see [13, Theorem 2.5]. For the third statement, one sees as for the first statement that C(X )⋊α ,r G is amenable. Then the conclusion follows from [2, Theorem 4.4.3]. The proofs of the first and third statements rely heavily on some nontrivial results for C∗-algebras. It is also possible to deduce the second statement in this fashion. As before, if ℓ1 (G) is amenable, then so is C∗r (G). Since G is discrete, [14, Theorem 4.2] then shows that G is amenable. Compared to this, Ringrose’s argument for the amenability of G is definitely more direct, and it would be desirable to have a more direct argument for the first and third statements as well. We do not know if, for non-trivial A, the amenability of ℓ1 (G, A; α ) implies the amenability of G. An attempt to derive such results via the enveloping C∗-algebra (as above) does not only seem overly complicated, but


17

may also be the wrong approach, since the passage to the enveloping C∗algebra simplifies the structure. For example, L1 (G) has a non-selfadjoint closed ideal for every abelian non-compact locally compact Hausdorff topological group (see [22, Theorem 7.7.1]), but this is of course no longer true for C∗ (G). In a similar vein, and directly related to amenability, we make the following observation in the non-discrete case. It is known (this goes back to [9]) that SL(2, ) is not amenable. Hence L1 (SL(2, )) is not amenable either. However, C∗ (SL(2, )) is amenable (see [4] for the amenability of every separable connected Lie group, and [15, p. 228] for the general possibly non-separable case). The information that the group is non-amenable is thus lost (or is at least no longer reflected in the amenability of the algebra) when passing from L1 (SL(2, )) to C∗ (SL(2, )). Such a lack of information in A ⋊α G can also occur for non-trivial A. Indeed, the Cuntz C∗-algebra O2 on two generators is amenable, but it was noted independently by Kumjian and Archbold (see [20, p. 83] for details) that O2 can be obtained as a C∗-algebra crossed product of the continuous functions on the Cantor set {0, 1}N by the non-amenable discrete group Z2 ∗ Z3 . Here, again, the information that G is not amenable is not present in A ⋊α G, or is at least not reflected in its amenability properties. It seems, therefore, at least conceivable that ℓ1 (G, A; α ) still contains enough information to be able to infer some kind of amenability property of G (or A) if ℓ1 (G, A; α ) itself is amenable, and it also seems that proper L1 -type arguments will have to be developed in order to investigate such Ringrose-type converses.

R

R

R

R

R

Acknowledgements. We thank Narutaka Ozawa and Jun Tomiyama for helpful comments. The last author was partially funded by FCT/Portugal through project UID/MAT/04459/2013. R EFERENCES [1] W.G. Bade, P.C. Curtis, and H.G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. (3) 55 (1987), no. 2, 359–377. [2] N.P. Brown and N. Ozawa, C∗ -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. [3] T. Ceccherini-Silberstein and M. Coornaert, Cellular automata and groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. [4] A. Connes, Classification of injective factors. Cases II1 , II∞ , IIIλ , λ 6= 1, Ann. of Math. (2) 104 (1976), no. 1, 73–115. [5] A. Connes, On the cohomology of operator algebras, J. Functional Analysis 28 (1978), no. 2, 248–253. [6] M. de Jeu, R. El Harti, and P.R. Pinto, Amenable crossed product Banach algebras associated with a class of C∗ -dynamical systems, Integral Equations Operator Theory 87 (2017), no. 2, 169–178.

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M ATHEMATICAL I NSTITUTE , L EIDEN U NIVERSITY, P.O. B OX 9512, 2300 RA L EIDEN , N ETHERLANDS E-mail address: [email protected]

THE

D EPARTMENT OF M ATHEMATICS AND C OMPUTER ENCES AND T ECHNIQUES , U NIVERSITY H ASSAN I, BP

S CIENCES , FACULTY OF S CI 577 S ETTAT , M OROCCO

E-mail address: [email protected] OF

D EPARTMENT OF M ATHEMATICS , I NSTITUTO S UPERIOR T ÉCNICO , U NIVERSITY L ISBON , AV. ROVISCO PAIS 1, 1049-001 L ISBON , P ORTUGAL E-mail address: [email protected]